On certain inequalities concerning the classical Euler’s gamma function

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Adv. Inequal. Appl. 2014, 2014:42 ISSN: 2050-7461

ON CERTAIN INEQUALITIES CONCERNING THE CLASSICAL EULER’S GAMMA FUNCTION

KWARA NANTOMAH

Department of Mathematics, University for Development Studies, Navrongo Campus,

P. O. Box 24, Navrongo, UE/R, Ghana.

Copyright c2014 Kwara Nantomah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.In this paper, some monotonic functions and some inequalities concerning certain ratios of generalized

gamma functions are established. The procedure utilizes the series forms of the generalized digamma functions.

Keywords:gamma function;q-gamma function;k-gamma function;(p,q)-gamma function; inequality.

2010 AMS Subject Classification:33B15, 26A48.

1. Introduction and Preliminaries

The classical Euler’s Gamma function,Γ(t)and the digamma function,ψ(t)are well-known in

literature as

Γ(t) =

Z ∞

0

e−xxt−1dx and ψ(t) = d

dt lnΓ(t) =

Γ0(t)

Γ(t),

t>0.

The p-Gamma function, Γp(t)and the p-digamma function,ψp(t)are defined for p∈N as

(see [6])

Γp(t) =

p!pt

t(t+1). . .(t+p) and ψp(t) =

d

dt lnΓp(t) =

Γ0_p(t) Γp(t)

, t >0.

Received September 2, 2014

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The q-Gamma function, Γq(t) and theq-digamma function, ψq(t) are also defined for q∈

(0,1)as (see [2])

Γq(t) = (1−q)1−t

∞

∏

n=1

1−qn

1−qt+n and ψq(t) =

d

dtlnΓq(t) =

Γ0_q(t) Γq(t)

, t>0.

Also, thek-Gamma function,Γk(t)and thek-digamma function,ψk(t)are defined fork>0

as (see [1])

Γk(t) =

Z ∞

0

e−xkk_xt−1_dx _and _ψ

k(t) =

d

dtlnΓk(t) =

Γ0_k(t) Γk(t)

, t>0.

In 2005, D´ıaz and Teruel [5] defined the(q,k)-Gamma and the(q,k)-digamma functions for

q∈(0,1)andk>0 as

Γ(q,k)(t) =

(1−qk)

t k−1

q,k

(1−q)tk−1

and ψ(q,k)(t) = d

dtlnΓ(q,k)(t) =

Γ0₍_q_,_k₎(t) Γ(q,k)(t)

, t>0,

where(t)_n,k=t(t+k)(t+2k). . .(t+ (n−1)k) =∏n_j₌−1₀(t+jk) is thek-generalized

Pochham-mer symbol.

Also in 2012, Krasniqi and Merovci [4] defined the(p,q)-Gamma and the (p,q)-digamma functions for p∈Nandq∈(0,1)as

Γ(p,q)(t) =

[p]t_q[p]q!

[t]q[t+1]q. . .[t+p]q

and ψ(p,q)(t) = d

dt lnΓ(p,q)(t) =

Γ0₍_p_,_q₎(t)

Γ₍_p_,_q₎(t), t >0.

where [p]q= 1−q

p

1−q.

The generalized digamma functions ψq(t), ψk(t), ψ(p,q)(t) and ψ(q,k)(t) as defined above

exhibit the following series representations (see also [8], [9], [10], [11] and [12]).

ψq(t) =−ln(1−q) +lnq

∞

∑

n=1

qnt

1−qn

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ψk(t) =

lnk−γ

k −

1

t +

∞

∑

n=1

t nk(nk+t) (2)

ψ₍_p_,_q₎(t) =ln[p]q+ (lnq) p

∑

n=1

qnt

1−qn

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ψ₍_q_,_k₎(t) = −ln(1−q)

k + (lnq)

∞

∑

n=1

qnkt

1−qnk

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In 2010, Krasniqi and Shabani [7] presented the following inequalities:

p−te−γt_Γ₍_α₎

Γp(α)

< Γ(α+t) Γp(α+t)

< p

1−t_eγ(1−t)_Γ₍_α₊₁₎

Γp(α+1)

fort∈(0,1), whereα is a positive real number such thatα+t >1.

Also in that same year, Krasniqi, Mansour and Shabani [6] presented the following results. (1−q)te−γt_Γ₍_α₎

Γq(α)

< Γ(α+t) Γq(α+t)

<(1−q)

t−1_eγ(1−t)_Γ₍_α₊₁₎

Γq(α+1)

fort∈(0,1), whereα is a positive real number such thatα+t >1 andq∈(0,1).

Several results of this nature as well as some generalizations have since been established. These can be found in the papers [8], [9], [10], [11] and [12].

In the present paper, our main objective is to present similar results involving the ratios

Γq(t) Γ(p,q)(t),

Γq(t) Γ(q,k)(t),

Γk(t)

Γ(p,q)(t) and Γk(t)

Γ(q,k)(t).

2. Auxiliary Results

Lemma 2.1. Letα >0, t>0, p∈N and q∈(0,1). Then,

ln(1−q) +ln[p]q+ψq(α+t)−ψ(p,q)(α+t)≤0.

Proof.By the series representations (1) and (3) we have, ln(1−q) +ln[p]q+ψq(t)−ψ(p,q)(t) = (lnq)

" ∞

∑

n=1

qnt

1−qn−

p

∑

n=1

qnt

1−qn

#

= (lnq)

∞

∑

n=p+1

qnt

1−qn ≤0.

Replacingtbyα+t completes the proof.

Lemma 2.2. Letα >0, t>0, q∈(0,1)and k≥1. Then,

ln(1−q)−1

kln(1−q) +ψq(α+t)−ψ(q,k)(α+t)≤0.

Proof.By the series representations (1) and (4) we have, ln(1−q)1−1k₊_ψ

q(t)−ψ(q,k)(t) = (lnq)

∞

∑

n=1

qnt

1−qn−

qnkt

1−qnk

≤0.

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Lemma 2.3. Letα >0, t>0, k>0, p∈N and q∈(0,1). Then,

ln[p]q−

lnk−γ

k +

1

α+t+ψk(α+t)−ψ(p,q)(α+t)>0.

Proof.By the series representations (2) and (3) we have, ln[p]_q−lnk−γ

k +

1

t +ψk(t)−ψ(p,q)(t) =

∞

∑

n=1

t

nk(nk+t)−(lnq)

p

∑

n=1

qnt

1−qn >0.

Lemma 2.4. Letα >0, t>0, q∈(0,1)and k>0. Then,

−ln(k(1−q))

k +

γ

k+

1

α+t +ψk(α+t)−ψ(q,k)(α+t)>0.

Proof.By the series representations (2) and (4) we have,

−ln(k(1−q))

k +

γ

k+

1

t +ψk(t)−ψ(q,k)(t) =

∞

∑

n=1

t

nk(nk+t)−(lnq)

∞

∑

n=1

qnkt

1−qnk >0.

3. Main Results

Theorem 3.1. Define a function S by

(5) S(t) = (1−q)

t

Γq(α+t)

[p]−qtΓ(p,q)(α+t)

, t∈(0,∞)

forα >0, p∈N and q∈(0,1). Then S is non-increasing on t∈(0,∞)and the inequalities

(6) (1−q)

−t

Γq(α)

[p]t

qΓ(p,q)(α)

≥ Γq(α+t)

Γ₍_p_,_q₎(α+t)≥

(1−q)1−tΓq(α+1)

[p]tq−1Γ(p,q)(α+1) are valid for every t∈(0,1).

Proof.Letg(t) =lnS(t)for everyt∈(0,∞). Then,

g(t) =ln(1−q)

t_Γ

q(α+t)

=tln(1−q) +tln[p]q+lnΓq(α+t)−lnΓ(p,q)(α+t). Then,

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That impliesgis non-increasing ont∈(0,∞). HenceSis non-increasing ont∈(0,∞)and for

everyt∈(0,1)we have,

S(0)≥S(t)≥S(1)

yielding the result.

Theorem 3.2. Define a function T by

(7) T(t) = (1−q)

t_Γ

q(α+t)

(1−q)kt_Γ

(q,k)(α+t)

, t∈(0,∞)

forα >0, q∈(0,1)and k≥1. Then T is non-increasing on t∈(0,∞)and the inequalities

(8) (1−q)

−t_Γ q(α)

(1−q)−kt_Γ

(q,k)(α)

≥ Γq(α+t)

Γ₍_q_,_k₎(α+t) ≥

(1−q)1−tΓq(α+1)

(1−q)1k(1−t)_Γ₍

q,k)(α+1) are valid for every t∈(0,1).

Proof.Leth(t) =lnT(t)for everyt∈(0,∞). Then,

h(t) =ln (1−q)

t_Γ

q(α+t)

(1−q)kt_Γ

(q,k)(α+t)

=tln(1−q)−t

kln(1−q) +lnΓq(α+t)−lnΓ(q,k)(α+t). Then, h0(t) =ln(1−q)−1

kln(1−q) +ψq(α+t)−ψ(q,k)(α+t)≤0. (by Lemma 2.2)

That implieshis non-increasing ont∈(0,∞). HenceT is non-increasing ont ∈(0,∞)and for

T(0)≥T(t)≥T(1)

establishing the result.

Theorem 3.3. Define a function U by

(9) U(t) = (α+t)k

−t ke

γt

k_Γ_k(_α+t)

, t∈(0,∞)

forα >0, p∈N, q∈(0,1)and k>0. Then U is increasing on t∈(0,∞)and the inequalities

(10) αk

t ke−

γt k_Γ_k(_α)

(α+t)[p]t_qΓ₍_p_,_q₎(α)<

Γk(α+t)

Γ₍_p_,_q₎(α+t)<

(α+1)k

t−1

k e

γ(1−t)

k _Γ_k(_α+1)

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Proof.Letµ(t) =lnU(t)for everyt∈(0,∞). Then,

µ(t) =ln(α+t)k

−t ke

γt

k_Γ_k(_α+t)

=ln(α+t)−t klnk+

γt

k +tln[p]q+lnΓk(α+t)−lnΓ(p,q)(α+t). Then,

µ0(t) =ln[p]q−

lnk−γ

k +

1

α+t+ψk(α+

t)−ψ₍_p_,_q₎(α+t)>0.(by Lemma 2.3)

That impliesµ is increasing ont∈(0,∞). As a result,U is also increasing ont∈(0,∞)and for

U(0)<U(t)<U(1)

yielding the result.

Theorem 3.4. Define a function V by

(11) V(t) = (α+t)e

γt

k_Γ_k(_α+t) ktk(1−q)

t k_Γ₍

q,k)(α+t)

, t ∈(0,∞)

forα >0, q∈(0,1)and k>0. Then V is increasing on t∈(0,∞)and the inequalities

(12) αk

t ke−

γt k_Γ_k(_α)

(α+t)(1−q)−

t k_Γ₍

q,k)(α)

< Γk(α+t) Γ₍_q_,_k₎(α+t)<

(α+1)k

t−1 k eγ

1−t

k _Γ_k(_α+1)

(α+t)(1−q)

1−t

k _Γ

(q,k)(α+1) are valid for every t∈(0,1).

Proof.Letδ(t) =lnV(t)for everyt∈(0,∞). Then,

δ(t) =ln (α+t)e

γt

k_Γ_k(_α+t) ktk(1−q)

t k_Γ

(q,k)(α+t)

=ln(α+t) +γt

k −

t

kln(k(1−q)) +lnΓk(α+t)−lnΓ(q,k)(α+t). Then,

δ0(t) =−ln(k(1−q))

k +

γ

k+

1

α+t+ψk(α+t)−ψ(q,k)(α+t)>0.(by Lemma 2.4)

That impliesδ is increasing ont∈(0,∞). As a result,V is also increasing ont∈(0,∞)and for

V(0)<V(t)<V(1)

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Conflict of Interests.

The author declares that there is no conflict of interests.

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